TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 37 



deduced from the approximate formula in accordance with the general 

 formula V, are in excellent agreement with the exact values. 



As illustrating the appropriate methods in the solution of problems 

 in electric circuit theory, it is of interest to derive the formula for the 

 band pass filter directly from the integral equation II. The method is 

 not only more generally applicable, but avoids the necessity of deriving 

 the definite integral (3.1). We therefore start with the formulas: 



f™e-P t A n (t)dt = l/pZ n (p) 



or 



f e-i> t A' n (t)dt = l/Z„(p), where A ' n (t)=d/ dtA„(t). 



For all wave-filters of the "ladder" type it may be shown that 



1 _ 1 (V5 Arrh - V^4)- 



Z n {p) z 2 \/r + r 2 Ji 



where Z\ and z 2 are the series and shunt impedances respectively, and 

 r=zi/z 2 . This expression admits of series expansion 



(2n+3) (2w+4) 1 



1 = 2_ T _1_ _ 2w+2 J. 



n (p) ~ ZxLr n 1! r n + l 



] 



1! r n+i ■ 2! r n + 2 



(2«+4) (2»+5) (2n+6) 1 



(3.22) 



3! r"+ 3 



For the L1C1L2C2 type of filter 



>Mf)W 



and 



It follows from (3.22) and the integral identity, 

 r e -P<A> n {t)dt = l/Z n (p) 

 that A n (t) has an expansion solution of the form 



+ (2« + 3)(2, + 4) ^ +4/ ^ 4(x) (323) 

 -< & + 4 H2»+5)(2»+6) p * 



3! 



